A231831 a(0) = 1; for n > 0, a(n) = -1 + 4*Product_{i=0..n-1} a(i)^2.

1, 3, 35, 44099, 85762231424099, 630794963141019085083178800095033630804099

Offset: 0

Author

Michel Marcus, Nov 14 2013

Keywords

• nonn

Comments

Sequence designed to show that there are an infinity of primes congruent to 3 modulo 4 (A002145). Terms are not necessarily prime. Their smallest prime factor from A002145 are: 3, 7, 11, 23, 4111, 2809343.

Next term is too large to include.

Similarly to Sylvester's sequence (A000058), it is unknown if all terms are squarefree (see also MathOverflow link). - Max Alekseyev, Mar 26 2023

Primes dividing terms of this sequence are listed in A362250. Since terms are pairwise coprime, for each n prime A362250(n) divides exactly one term, whose index is A362251(n). That is, A362250(n) divides a(A362251(n)). - Max Alekseyev, Apr 16 2023

Links

• S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.
• fredrickmnelson et al., Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?, MathOverflow, 2023.

Crossrefs

Cf. A000058, A002145, A007018, A231830, A362250, A362251.

Programs

PARI

lista(nn) = {a = vector(nn); a[1] = 3; for (n=2, nn, a[n] = 4*prod(i=1, n-1, a[i]^2) - 1;); a;}

Formula

For n > 1, a(n) = (a(n-1) + 1) * a(n-1)^2 - 1. - Max Alekseyev, Mar 26 2023

Extensions

a(0) = 1 prepended by Max Alekseyev, Mar 26 2023